A256209 Coefficients of mock modular form H_2^(4) (divided by 16).

1, 3, 7, 14, 27, 49, 84, 141, 230, 364, 567, 867, 1302, 1932, 2829, 4091, 5859, 8309, 11675, 16275, 22513, 30914, 42174, 57176, 77049, 103263, 137669, 182616, 241110, 316910, 414750, 540603, 701903, 907928, 1170261, 1503238, 1924607, 2456349

Offset

0,2

Comments

The coefficients occur on page 94, Table 24, column 1A for McKay-Thompson series H_{1A,2}^(4) in the Cheng et al. arXiv article. - Michael Somos, Nov 04 2015

References

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 3, 2nd equation.

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Miranda C. N. Cheng, John F. R. Duncan, Jeffrey A. Harvey, Umbral Moonshine, arXiv:1204.2779 [math.RT], 2012-2013.

Formula

G.f.: Sum_{k>0} x^(k-1) * (1 + x) * ... * (1 + x^(2*k-2)) / ((1 + x) * (1 + x^3) * ... (1 + x^(2*k-1)))^2. - Michael Somos, Nov 04 2015

2 * a(n) = A053270(3*n) - A257640(3*n). - Michael Somos, Nov 04 2015

Example

G.f. = 1 + 3*x + 7*x^2 + 14*x^3 + 27*x^4 + 49*x^5 + 84*x^6 + 141*x^7 + ...
G.f. = q^3 + 3*q^7 + 7*q^11 + 14*q^15 + 27*q^19 + 49*q^23 + 84*q^27 + ...

Mathematica

nmax = 50; a:= CoefficientList[Series[q*Sum[q^(k - 1)*(Product[1 + q^j, {j, 1, 2 k - 2}])/(Product[1 - q^(2 j - 1), {j, 1, k}])^2, {k, 0, nmax}], {q, 0, 150}], q]; Table[a[[n]], {n, 1, 100}] (* G. C. Greubel, Jul 27 2018 *)

Prog

(PARI) {a(n) = if( n<0, 0, n++; polcoeff( sum(k=1, n, x^k * prod(i=1, 2*k - 2, 1 + x^i, 1 + x * O(x^(n - k))) / prod(i=1, k, 1 - x^(2*i - 1), 1 + x * O(x^(n - k)))^2), n))}; /* Michael Somos, Nov 04 2015 */

Crossrefs

Equals A256052/8.

Cf. A053270, A257640.

Sequence in context: A276024 A274233 A369576 * A236914 A152902 A027084

Adjacent sequences: A256206 A256207 A256208 * A256210 A256211 A256212

Keyword

nonn

Author

N. J. A. Sloane, Mar 25 2015

Status

approved